Formation enthalpies by mixing GGA and GGA $+$ $U$ calculations

Anubhav Jain, Geoffroy Hautier, Shyue Ping Ong, Charles J. Moore, Christopher C. Fischer, Kristin A. Persson, and Gerbrand Ceder

Phys. Rev. B 84, 045115 – Published 12 July 2011

Abstract

Standard approximations to the density functional theory exchange-correlation functional have been extraordinary successful, but calculating formation enthalpies of reactions involving compounds with both localized and delocalized electronic states remains challenging. In this work we examine the shortcomings of the generalized gradient approximation (GGA) and GGA $+$ $U$ in accurately characterizing such difficult reactions. We then outline a methodology that mixes GGA and GGA $+$ $U$ total energies (using known binary formation data for calibration) to more accurately predict formation enthalpies. We demonstrate that for a test set of 49 ternary oxides, our methodology can reduce the mean absolute relative error in calculated formation enthalpies from approximately 7.7–21% in GGA $+$ $U$ to under 2%. As another example we show that neither GGA nor GGA $+$ $U$ alone accurately reproduces the Fe-P-O phase diagram; however, our mixed methodology successfully predicts all known phases as stable by naturally stitching together GGA and GGA $+$ $U$ results. As a final example we demonstrate how our technique can be applied to the calculation of the Li-conversion voltage of LiFeF $_{3}$ . Our results indicate that mixing energies of several functionals represents one avenue to improve the accuracy of total energy computations without affecting the cost of calculation.

Received 13 April 2011

DOI:https://doi.org/10.1103/PhysRevB.84.045115

Authors & Affiliations

Anubhav Jain, Geoffroy Hautier, Shyue Ping Ong, Charles J. Moore, Christopher C. Fischer, Kristin A. Persson, and Gerbrand Ceder^*

Massachusetts Institute of Technology, 77 Massachusetts Avenue 13-5056, Cambridge, Massachusetts 02139, USA

^*Corresponding author: gceder@mit.edu

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Vol. 84, Iss. 4 — 15 July 2011

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Figure 1
Calculation methodology for mixing GGA and GGA $+$ $U$ calculations. The elements and all delocalized/uncorrelated compounds are computed with GGA. We compute with GGA $+$ $U$ all localized compounds; in this work, this includes any oxide containing one of the elements listed in Table . Other types of localized compounds, such as transition metal sulfides or fluorides, might also fall within the realm of GGA $+$ $U$ calculation. The energy adjustment applied to GGA $+$ $U$ can be envisioned as a virtual reaction between the elements and localized binary compounds that is determined using experimental data. This energy adjustment is described in Eq. (6); details are provided in the surrounding text.Reuse & Permissions
Figure 2
Differences between experimental and calculated and formation enthalpies as a function of metal content for V, Cr, Mn, Fe, Co, Ni, Cu, and Mo. The calculations used GGA for the metals and GGA $+$ $U$ for the binary and ternary oxides. The differences from experiment increase linearly with metal content in the oxides; this difference can be removed simply by adjusting GGA $+$ $U$ energies for the oxides via Eq. (6). The gray line is fitted via least-squares to the binary-oxide data alone (red squares), and its slope represents the magnitude of the adjustment Δ $E_{M}$ , which is specific to the metal and $U$ value. However, ternary oxides are also well represented by this line, suggesting that the adjustment generalizes beyond the binary oxides.Reuse & Permissions
Figure 3
Computed and experimental formation enthalpies of ternary oxides from the elements using (a) GGA $+$ $U$ for all compounds with Wang's $O_{2}$ correction, (b) GGA $+$ $U$ with an $O_{2}$ correction fitted to minimize the MARE, and (c) the mixed GGA/GGA $+$ $U$ framework proposed in this paper. The mixed GGA/GGA $+$ $U$ framework best reproduces experimental results while retaining an $O_{2}$ energy consistent with GGA calculations. The full list of compounds and results can be found in the Appendix.Reuse & Permissions
Figure 4
Strategy for generating the Fe-P-O phase diagram using a mixture of GGA and GGA $+$ $U$ calculations. Regions of the phase diagram containing both Fe and O are computed with GGA $+$ U, whereas the remaining regions are computed with GGA. This strategy follows the more general guidelines outlined in Fig. 1. The $O_{2}$ energy is taken from Wang et al.'s fit.[15]Reuse & Permissions
Figure 5
Computed 0 K, 0 atm phase diagram of the Fe-P-O system within the (a) GGA methodology, (b) GGA $+$ $U$ methodology, and (c) mixed GGA/GGA $+$ $U$ technique of the current work. The GGA phase diagram fails to reproduce known phases in the Fe-O and Fe-P-O composition ranges, whereas the GGA $+$ $U$ methodology fails along the Fe-P line. Only the mixed GGA/GGA $+$ $U$ phase diagram of the current work successfully reproduces known phases in all composition areas.Reuse & Permissions

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